Solve frac{3x^{2}y}{4x^3y^5}div6y^2/2xy^3

$$\frac { 3 x ^ { 2 } y } { 4 x ^ { 3 } y ^ { 5 } } \div \frac { 6 y ^ { 2 } } { 2 x y ^ { 3 } }$$

Evaluate

$$\frac{1}{4y^{3}}$$

Solution Steps

Divide \frac{3x^{2}y}{4x^{3}y^{5}} by \frac{6y^{2}}{2xy^{3}} by multiplying \frac{3x^{2}y}{4x^{3}y^{5}} by the reciprocal of \frac{6y^{2}}{2xy^{3}}.

$$\frac{3x^{2}y\times 2xy^{3}}{4x^{3}y^{5}\times 6y^{2}}$$

Cancel out 2\times 3xyx^{2}y^{3} in both numerator and denominator.

$$\frac{1}{2\times 2yy^{2}}$$

To multiply powers of the same base, add their exponents. Add 1 and 2 to get 3.

$$\frac{1}{2\times 2y^{3}}$$

Multiply 2 and 2 to get 4.

$$\frac{1}{4y^{3}}$$

Differentiate w.r.t. y

$$-\frac{3}{4y^{4}}$$

Steps Using Chain Rule

Divide \frac{3x^{2}y}{4x^{3}y^{5}} by \frac{6y^{2}}{2xy^{3}} by multiplying \frac{3x^{2}y}{4x^{3}y^{5}} by the reciprocal of \frac{6y^{2}}{2xy^{3}}.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{3x^{2}y\times 2xy^{3}}{4x^{3}y^{5}\times 6y^{2}})$$

Cancel out 2\times 3xyx^{2}y^{3} in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{1}{2\times 2yy^{2}})$$

To multiply powers of the same base, add their exponents. Add 1 and 2 to get 3.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{1}{2\times 2y^{3}})$$

Multiply 2 and 2 to get 4.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{1}{4y^{3}})$$

If F is the composition of two differentiable functions f\left(u\right) and u=g\left(x\right), that is, if F\left(x\right)=f\left(g\left(x\right)\right), then the derivative of F is the derivative of f with respect to u times the derivative of g with respect to x, that is, \frac{\mathrm{d}}{\mathrm{d}x}(F)\left(x\right)=\frac{\mathrm{d}}{\mathrm{d}x}(f)\left(g\left(x\right)\right)\frac{\mathrm{d}}{\mathrm{d}x}(g)\left(x\right).

$$-\left(4y^{3}\right)^{-1-1}\frac{\mathrm{d}}{\mathrm{d}y}(4y^{3})$$

The derivative of a polynomial is the sum of the derivatives of its terms. The derivative of a constant term is 0. The derivative of ax^{n} is nax^{n-1}.

$$-\left(4y^{3}\right)^{-2}\times 3\times 4y^{3-1}$$

Simplify.

$$-12y^{2}\times \left(4y^{3}\right)^{-2}$$